Nonvanishing section of infinite-dimensional tautological bundle II

Question

This is a follow-up question on a previous question of mine, which ended up to be trivial, because I overlooked the obvious problem with Hilbert space bundles, which I fix here.

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Let us write $E$ for the space of complex-valued sequences $\{\dots, a_{-2}, a_{-1}, a_1, a_2, \dots\}$, labeled by $(\mathbb{Z}\setminus\{0\})$, such that only finitely many $a_n$ are non-zero. For a subset $I \subset \mathbb{Z}$, write $$E_I = \mathrm{span}\{ e^i ~|~ i \in I\} \subseteq E,$$ where $e^i$, $i \in \mathbb{Z}$ are the canonical basis vectors of $E$.

Let us call a vector subspace $W \subset E$ half-dimensional if there exists $N \in \mathbb{N}$, such that $$ W = W^\prime \oplus E_{[N, \infty)},$$ where $W^\prime$ is a half-dimensional subspace of the (finite-dimensional) space $E_{[-N, N]}$. Consider the following space $$\mathrm{Gr}_\infty = \{ W \subset E ~|~ W \text{ is half-dimensional} \}.$$ This has a natural direct limit topology: $ \mathrm{Gr}_\infty = \lim_{n \to \infty} \mathrm{Gr}_n(E_{[-n, n]}),$ where the map $\mathrm{Gr}_n(E_{[-n, n]}) \to \mathrm{Gr}_{n+1}(E_{[-(n+1), n+1]})$ is given by sending $W \to W \oplus \mathbb{C} e^{n+1}$.

There is a tautological bundle $\tau$ over $\mathrm{Gr}_\infty$, with fibers $\tau(W) = W$.

Q: Does this bundle have a non-vanishing section?

In the previous version of the question, the answer was trivially yes: the bundle was a bundle of Hilbert spaces, which has contractible automorphism group. Here, we have a bundle with typical fiber $\mathbb{C}^\infty$, the automorphism group of which is $U_\infty = \lim_{n \to \infty} U_n$, which has lots of trivial topology.

Answer 1

$\DeclareMathOperator{\Gr}{Gr}\newcommand{\C}{\mathbb{C}}\DeclareMathOperator*{\colim}{colim}$The total space of your bundle is the colimit of the spaces $\tau_n\times\C^{[N,\infty)}$, where $\tau_n\to \Gr_n(E_{[-n,n]})$ is the tautological bundle over $\Gr_n(E_{[-n,n]})$ and the transition functions are the product of the bundle map $\tau_n\oplus \C\to \tau_{n+1}$, which exhibits the left-hand side as the pullback of $\tau_{n+1}$ along the inclusion $\Gr_n(E_{[-n,n]})\hookrightarrow \Gr_{n+1}(E_{[-(n+1),n+1]})$, and the identity of $\C^{[N+1,\infty)}$. In particular, it receives a map from $\colim_n \tau_n$, and the induced map from the complements of zero sections $\colim_n \tau_n^\times$ lands in the complement of the zero section. Now the map $\tau_n^\times\to \Gr_n(E_{[-n,n]})$ is a fiber bundle, in particular a Serre fibration, with fiber $S^{2n-1}$. Since a filtered colimit of Serre fibrations is a Serre fibration, the map $\colim_n \tau_n^\times\to \Gr_\infty$ is a Serre fibration with fiber $\colim_n S^{2n-1}\simeq *$, i.e. an acyclic Serre fibration. Since the base $\Gr_\infty$ has the homotopy type of a CW complex, there is a section $\Gr_\infty\to\colim_n\tau_n^\times$, which then maps to a nonzero section of your bundle.

It is probably also possible to explicitly construct such a section by using that the shift map on sequences is homotopic to the identity.